376 MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OP NUMBERS. 
\«i 
1 - . 1 - (f /3.x.,)"'" . 1 - (y.x., 
x «2 
7!, + l 
to nil factors 
l-(hi^X,i) .1 
ao 
?ii +1 
h A V 1 ( h y-2 ^ 4- r 4- 
— Aoo . 1 — r;- Ao 3 to w, lactors 
h Pi / V »2 7 l / 
1 ( Cl ^ X 31 
1 _ I A |i . 1 _ -2i xA ”" to m, factors 
Cl ^2 ) \ C-l 72 / 
&C. 
to /i factors 
to U factors 
to /, factors 
a. 
1 — aittiXii . 1 — — AX 12 • 1 --^ yiXi 3 . . . to Wi factors 
a 
1 — Xoi. 1 — Xgo. 1 — Y- — X 03 ... to Wo factors 
«i ■■ \ Pi ^ h 7i 
1 - Cl ^ X 31 . 1 
“ X 30 . 1 — — X 33 ... to W 3 factors 
Cl ^0 C2 70 
to /i factors to 4 factors to /s factors 
&c. 
wherein, naturally, each of the series 
4, 
to 
4) • • 
Wi, 
Wo, 
. . 
?ll. 
w. 
na, . . 
is in descending order, and the theorem of reciprocity involved in the fact of the 
existence of the graph consists in the circumstance that the function remains 
unaltered, when X^j is put equal to x, for any substitution impressed upon the unsuffixed 
symbols I, ni, n. 
In the corresponding lattice the conditions are :— 
(i.) The first, second, &c,, rows do not contain more than n 2 , &c. numbers 
respectively ; 
(ii.) The first, second, &c., rows do not contain higher numbers than 4, L, &c. . . . ; 
(hi.) No number so great as s occurs below row w^ for all values of s; 
wii, m 2 , . . . w^ . . . being of course in descending order of magnitude. 
Art. 93. The reduction of this O function presents great difficulties, and I propose 
to restrict consideration to the case 
4 = 4 = I, = ... = l 
my = m 2 = W 3 = ... — m 
n. 
